TITLE: Why is $GL(B)$ a Banach Lie Group?
QUESTION [9 upvotes]: Banach Lie Groups are what you'd expect: 
https://www.encyclopediaofmath.org/index.php/Lie_group,_Banach
If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a Banach Lie group?

REPLY [9 votes]: This follows from three simple observations:

The subset $\operatorname{GL}(B) \subset B$ is open, so it is a Banach manifold modeled on $B$ itself.
Multiplication is the restriction of a continuous linear map, hence it is analytic.
Inversion is locally given by the Neumann series, hence it is analytic, too.